The Meaning of Einstein's Equation

TL;DR

This paper provides an accessible introduction to general relativity, focusing on the geometric interpretation of Einstein's equation through the motion of free-falling particles, suitable for students and teachers.

Contribution

It offers a simplified formulation of Einstein's equation based on particle motion, clarifying its geometric meaning and including an annotated bibliography for further study.

Findings

Simplified geometric formulation of Einstein's equation

Explanation of equivalence to tensor formulation

Annotated bibliography for educational resources

Abstract

This is a brief introduction to general relativity, designed for both students and teachers of the subject. While there are many excellent expositions of general relativity, few adequately explain the geometrical meaning of the basic equation of the theory: Einstein's equation. Here we give a simple formulation of this equation in terms of the motion of freely falling test particles. We also sketch some of its consequences, and explain how the formulation given here is equivalent to the usual one in terms of tensors. Finally, we include an annotated bibliography of books, articles and websites suitable for the student of relativity.